Abstract: The kernel of Cauchy integrals is expanded in an asymptotic expansion of Chebyshef polynomials, and then the principal value of Cauchy integrals is calculated by integrating it into a close form term by term and summing the finite terms. Based on this, distinguishing the different types, by means of analytic differentiation term by term or cubic spline derivative with respect to the principal value of Cauchy integrals obtained, the principal value of Mangier is calculated. It will not restrict the positions of calculated points and the integral kernel, except the entirely natural condition that the kernel is " quadratical integrable". With present method the principal values of Cauchy and Mangier integrals of highly oscillatory kind can be treated. Eight numerical examples are given. The comparison between the calculated results of some exact solutions and the author's method shows clearly, that the present algorithm is of fine accuracy.

Key words: numerical calculation, numerical integration, Chebyshev polynomial